Existence and uniqueness of a nontrivial solution for a class of third-order nonlinear p -Laplacian m -point eigenvalue problems on time scales

Abstract

In this paper, we study the existence and uniqueness of a nontrivial solution for the following third-order p -Laplacian m -point eigenvalue problems on time scales: { ( ϕ p ( u Δ ∇ ) ) ∇ + λ f ( t , u ( t ) , u Δ ( t ) ) = 0 , t ∈ ( 0 , T ) , α u ( 0 ) − β u Δ ( 0 ) = 0 , u ( T ) = ∑ i = 1 m − 2 a i u ( ξ i ) , u Δ ∇ ( 0 ) = 0 , where ϕ p ( s ) is p -Laplacian operator, i.e., ϕ p ( s ) = | s | p − 2 s , p > 1 , ϕ p − 1 = ϕ q , 1 p + 1 q = 1 , λ > 0 is a parameter, 0 < ξ 1 < ⋯ < ξ m − 2 < ρ ( T ) . We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when λ is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given. The conditions we used in the paper are different from those in [Z.C. Hao, L. Debnath, On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems, Appl. Math. Lett. 18 (2005) 543–553; D. Jiang, R.P. Agarwal, A uniqueness and existence theorem for a singular third-order boundary value problem on [ 0 , + 1 ) , Appl. Math. Lett. 15 (2002) 445–451; S.H. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem, J. Math. Anal. Appl. 323 (2006) 413–425; Z. Liu, J.S. Ume, D.R. Anderson, S.M. Kang, Twin monotone positive solutions to a singular nonlinear third-order differential equation, J. Math. Anal. Appl. 334 (2007) 299–313; Q. Yao, Positive solutions to a special singular second-order boundary value problem, Math. Comput. Model. (in press); Q. Yao, Successive iteration of positive solution for a discontinuous third-order boundary value problem, Comput. Math. Appl. 53 (2007) 741–749].